1. **Stating the problem:** We have a horizontal line segment AB of length 8 cm. Point P is located 3 cm from B on segment AB. From point P, a vertical line segment PQ of length 4 cm is drawn perpendicular to AB. 2. **Goal:** Find the length of segment AP. 3. **Understanding the setup:** Since AB is 8 cm and P is 3 cm from B, the length PB = 3 cm. Therefore, AP = AB - PB. 4. **Calculation:** $$AP = 8 - 3$$ $$AP = 5$$ 5. **Summary:** The length of segment AP is 5 cm. 6. **Additional info:** The vertical segment PQ is 4 cm, but it does not affect the length of AP along AB. **Final answer:** $$\boxed{5\text{ cm}}$$